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Stochastic analysis provides the mathematical framework for understanding systems that evolve continuously under the influence of randomness.

From stochastic equations to modern applications

From the erratic motion of pollen grains suspended in water, first documented by botanist Robert Brown in 1827, to the fluctuating prices of financial assets, such random continuous motion is ubiquitous in nature and society. The rigorous study of such phenomena, led Kiyoshi Itô in the 1940s, to develop the theory of stochastic integration and stochastic differential equations (SDEs), which describe how a system changes continuously, driven at each instant by a random perturbation. This framework underpins, for example, the celebrated Black–Scholes model for option pricing, and remains central to modern mathematical finance.

SPDEs, noise, and computation

More recently, when the spatial extent of a system matters, researchers have turned to stochastic partial differential equations (SPDEs), which model phenomena as varied as fluctuating interfaces, phase separation, and large systems of interacting particles. When the driving noise is very rough, new mathematical frameworks are required to make rigorous sense of the equations, notably the theory of regularity structures, for which Martin Hairer was awarded a Fields Medal in 2014.

Alongside these theoretical developments, there is a growing need for efficient numerical algorithms capable of approximating solutions to stochastic equations, with applications ranging from nuclear physics to machine learning.

Members of ProbLaB with research in these areas include:

• , Professor; works on optimal stopping, the Skorokhod embedding problem, and martingale optimal transport, with a particular interest in robust and model-free approaches to the pricing and hedging of financial derivatives
• , Lecturer; whose research sits at the intersection of stochastic analysis, PDE theory, and numerical analysis, and who studies equations describing the stochastic evolution of particle densities as limits of large interacting particle systems, with connections to machine learning
• , Senior Lecturer; develops high-order numerical methods for SDEs drawing on ideas from rough path theory, and applies these to prominent equations in data science such as Langevin dynamics
• , Lecturer; studies singular SPDEs and the phenomenon of regularisation by noise — whereby adding randomness to an ill-behaved deterministic equation can paradoxically restore good mathematical properties — as well as McKean–Vlasov equations arising as limits of large interacting particle systems, with applications to models from mathematical biology and statistical physics
• , Professor; investigates how large-scale collective behaviour — such as pattern formation, synchronisation, and epidemic spreading — emerges from the random interactions of many individuals, drawing on connections between stochastic processes, kinetic theory, and applications in ecology and the social sciences